1 tablespoon olive oil
1 medium yellow onion, diced
4 medium carrots
3 medium celery stalks
8 cups (2 quarts) water
1/2 teaspoon dried thyme
2 bay leaves
1 (4- to 5-pound) whole chicken, neck and giblets removed from the cavity
1 tablespoon kosher salt, plus more as needed
1/2 teaspoon freshly ground black pepper, plus more as needed
2 tablespoons coarsely chopped fresh Italian parsley leaves
2 cups cooked rice or noodles (optional)
Heat the oil in a medium frying pan over medium heat until shimmering. Add the onion and cook, stirring occasionally, until softened and starting to brown, about 7 minutes. Meanwhile, peel and cut the carrots into large dice, then cut the celery into large dice. Transfer both to a 6-quart or larger slow cooker. Add the water, thyme, and bay leaves. When the onion is ready, transfer to the slow cooker and stir to combine.
Place the chicken on a work surface or cutting board and pat dry with paper towels. Season inside and out with the salt and pepper. Place the chicken on top of the vegetables in the slow cooker. Cover and cook until the chicken is cooked through and falling off the bones, about 6 hours on the HIGH setting or 8 hours on the LOW setting.
Transfer the chicken to a rimmed baking sheet (it’s okay if it falls apart into large pieces). When cool enough to handle but still warm, use your hands to shred the meat into bite-sized pieces (discard the skin, cartilage, and bones). Return the shredded chicken to the slow cooker and add the parsley and a few large pinches of salt.
Add the rice or noodles, if using, and stir to combine. Taste and season with salt and pepper as needed.
Make ahead: The onion can be sautéed and refrigerated in an airtight container for up to 2 days.